Because Alliance killing him would only cause more agression from the Horde.

Bare it with me as I explain:

Throughout History, alliance was always known on stepping on Horde's toes, destruction of Dark Portal and defeat at Blackrock Mountain was the final blow that shatterd them and later put into camps.
Than the big 3 came (Thrall, Hellscream and Orgrimm) and liberated their people, but Warchief Orgrimm Doomhammer dies at the liberation of Hammerfall, killed my some non-important human, reducing the morality of orcs even more.
And now they're forced to hide in the forests from Humans of Lorderon, because they know they'll likely be killed on site, since they're known as the kind of monsters that prefer to go down fighting rather than taken prisoners.

So Thrall finally leads the Horde across the sea to Kalimdor, which remindes them much of their homeland.
But Jaina, leading the survivours of Lorderon left for Kalimdor as well at the urge of Medhiv, and once again, orcs see it as if the Humans followed them to hunt them down.

Aftermath the Battle for Mount Hyjal, the Horde finally settles down in the wastes and starts building a city which they could call their home, but the zelous humans of Kul Tiras, lead by Jaina's father, want to eradicate them all, simply because of "ORCS ARE EVIL! THEY WILL NEVER CHANGE! THEY MUst DIE LIKE THE MONSTERS THEY ARE!" thinking, where as orcs just wanted to be left alone and have a place to call home on an alien world they were stranded on.

And than we get lots of crap which Varian caused before and during WotLK, like attacking Thrall during the Theramore Peace Summit, where Twilight Hammers made a tactical ambush made from Horde races, attacking Thrall during Battle for Undercity claiming "what would our world be without YOUR Horde", and than the Garrosh/Varian clash during Ulduar thing (tho who's really to blame, either Varian who started the provocation or Garrosh for being easly provoken).

Camp Taurajo however seemed to be the big nail to the coffin.

Either way, what I'm getting at is that everytime Horde (or orcs) try to mind their own buisness, the Alliance (or Humans specifically) have to ruin it for them, the whole race-war thing.
Now don't get started on the whole "but horde did it 1st, they deserve to get it in return!".
As Tharan Zhu explain, the reason why Allaince and Horde can never have peace is, because each time one tries to get revange on another, it is an act of agression in of itself, and than that same agression triggers the lust for revange in opposite faction; thus creating the endless cycle of revange.

Now here's the scenario:

Horde rebels and ALLIANCE assault Garrosh and his followers in HORDE capital city. That mite be acceptable to rebes of Horde, because they asked for their help, because they know that with power that Garrosh is amassing in Orgrimmar isn't something that one side can defeat on their own.

But the deciding point will go to, who will deal with Garrosh in the end (assuming he survives the whole thing, what with Heart of Old God being there)?

Both sides have every right to do him in, however if the Alliance decides to take him out, it will be seen as an act of agression. YES, Garrosh is a merciless tyrant and he deserves to die, but he's also an ORC and a war veteran of Northrend Campagin, so his execution at the hand of opposite faction will be like saying to them "WE KILLED HIM BECAUSE HE WAS AN ORC."
I mean how would you feel if a foreign force would come into your territory and judge the leader of your people by THIER standards instead of your own?
You probaly wouldn't be happy either.

So if anything, Garrosh MUST be dealt with at the hands of his own people. ONLY THAN will justice be met.

If you recall, it was announced a long time ago, before Cata was even over, that we were finally going to get to kill numb-nuts. This was in response to all the Alliance bitching, and rightfully so, that the story had turned so Horde Centric. It was supposed to be the "bone" that got thrown to the Alliance for having to suffer through that abomination of a story called cata which focused so heavily around Green Jesus. Not to mention all the other shortfalls in Alliance gameplay and story, hello Twilight Highlands. (sorry we ran out of time. - I mean really... wtf...)

I had to suffer through that terrible quest-line of saving Thrall. All the while, not slitting the throat of his whiny ass bitch of a fuck-buddy who has for the entire history of the game been as anti-alliance as you can get.

Alliance players deserve the kill. Hell, we were PROMISED the kill. The Horde can watch a cut-screen of it from the view of a crashing Airship that's driven by an annoying and drunk goblin.

I sat alone in the dark one night, tuning in by remote.
I found a preacher who spoke of the light, but there was Brimstone in his throat.
He'd show me the way, according to him, in return for my personal check.
I flipped my channel back to CNN and lit another cigarette.

I'd have to agree with the idea that the Horde taking him out carries more weight. The alliance certainly has every reason to want to remove Garrosh, but that would still boil down to "the main guys of our side get to kill the main guy from the other side" conflict that tends to make up faction wars. I think it means more, to the world/lore of warcraft, to have the citizens and leaders of the Horde ultimately be the ones to remove Garrosh from power.

To put it another way, Garrosh is certainly both factions problem, but he is primarily the Horde's responsibility. Especially with Thrall's involvement in the 5.3 rebellion quests and very likely the raid. Garrosh leading the Horde is literally Thrall's responsibility, it makes the most sense for him and Vol'Jin to be the ones who take Garrosh down.

I play and love both sides, but started out as an Alliance Player. Personally I feel the Horde deserves the kill, as its their civil war. They are broken, torn between two sides, and every race has felt Garrosh's lashes.

I do believe that if the end battle have us fighting alongside both alliance and horde npcs, Varian will retreat his forces once Garrosh is defeated, and leave it to the other leaders of the Horde to punish the warchief for his actions and crimes.

Well the alliance is, apparently, coming together as a faction and becoming stronger, its the horde that is breaking apart.

Well the alliance has good reason to want to kill Garrosh, mostly based on vengeance, its the horde that benefits most from the kill, because its not just them taking down a tyrant in there own city, but also showing there prepared to finally say enough is enough, we're tired of your crap, we don't care if your our leader.

The only question would remain which lore character gets the kill. Whoever it is, both factions will be bitching and complaining about whatever lore character it is getting a kill steal.

---------- Post added 2013-04-18 at 11:48 PM ----------

Originally Posted by GennGreymane

the alliance because to hell with the horde thats why

Vol'jin's leading a revolution to get rid of Garrosh.. haven't seen genn do much to retake Gilneas.

Well the alliance is, apparently, coming together as a faction and becoming stronger, its the horde that is breaking apart.

Well the alliance has good reason to want to kill Garrosh, mostly based on vengeance, its the horde that benefits most from the kill, because its not just them taking down a tyrant in there own city, but also showing there prepared to finally say enough is enough, we're tired of your crap, we don't care if your our leader.

The only question would remain which lore character gets the kill. Whoever it is, both factions will be bitching and complaining about whatever lore character it is getting a kill steal.

Now that I think of it, the Horde hasn't challenged and overthrown its leader since the incident with Doomhammer and Blackhand.

Now that I think of it, the Horde hasn't challenged and overthrown its leader since the incident with Doomhammer and Blackhand.

Also, the Horde (or actually mostly the orcs) haven't been this brainwashed since Ner'zhul and Gul'dan had them slaughter the Draenei. Now too, as then, the Orcs are pursuing dark magics (dark shaman) and doing horrible things to both people and land alike. If you've read The Shattering and Tides of war it basically tells of how the 'people' look up to Garrosh for his heroic deeds, and are also impressed with his victories over the Alliance. They are feeling victorious again, and therefore treading down a familiar and dark path.

Vol'jin's leading a revolution to get rid of Garrosh.. haven't seen genn do much to retake Gilneas.

This may be true but sadly Blizzard does not allow him to act. He is portrayed as headstrong and ferocious with great fighting spirit in the book. It is totally illogical for him to remain as passive as he is shown in the game.
Hopefully this will change at the end of MoP and the Gilneas issue will be adressed. (... yeah right... )